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The phrase "a criterion on which" is correct and usable in written English.
It can be used when discussing a standard or principle that serves as a basis for judgment or evaluation.
Example: "The success of the project will be measured by a criterion on which all team members agree."
Alternatives: "a standard by which" or "a basis for".
Exact(3)
Although none of the estimates had p-values below the Benjamini-Krieger-Yekutieli corrected critical value, we wonder whether this may be too harsh a criterion on which to judge these estimates.
As pre-existing chronic health conditions may be construed as a disability under the UK Equality Act 2010 [ 20], a past history of depression or an eating disorder cannot lawfully be a criterion on which to refuse entry per se.
DKA may also occur during acute intercurrent illness and should not be used as a criterion on which to base the diagnosis of type 1 versus type 2 diabetes (10).
Similar(57)
But statistically, love is not a sound criterion on which to base a relationship; after all, a 50/50 chance of success is no better than flipping a coin.
Arthrogryposis, which was observed in 94% of the infected sheep flocks, 82% of the infected cattle herds, and 81% of the infected goat flocks, appears to be a sensitive criterion on which to form a clinical suspicion of congenital SBV disease in ruminants.
That doesn't mean that all men outperform all women, but it means that for élite-sport discussion, that Y chromosome, and specifically the SRY gene on it, which directs the formation of testes and the production of testosterone, is a key criterion on which to separate people into categories...
It is known that the extent of bivalent formation during meiotic prophase in polyploid plants rarely depends on chromosome homology alone, and therefore meiotic configuration may be a weak criterion on which to base genomic relationships (de Wet and Harlan, 1972; Jauhar and Joppa, 1996; Kopecký et al., 2008).
Additionally SWISS scores may not represent the only criterion on which one method is preferred.
The main strength of our study is that it provides a comparison with one of the most recognized instruments for neurodevelopmental assessment, the Griffiths, which gives a standardized independent criterion on which performances at the Bayley Scales can be referred.
See Norman (1995) for the idea that negatives with *p- initial and negative *m- initial are an important grammatical criterion on which northern and southern Chinese dialects can be distinguished.
What arguably justifies such a comparison, though, is the absence of a transitivity corpus study of the experience of a social actor other than the terminal patient that could be used as a criterion based on which the results of this study could be judged.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com